Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:

import java.io.*;
import java.util.*;
public class Main{
	static BufferedReader br;
	static int cin() throws Exception
	{
		return Integer.valueOf(br.readLine());
	}
	static int[] split() throws Exception
	{
		String[] cmd=br.readLine().split(" ");
		int[] ans=new int[cmd.length];
		for(int i=0;i<cmd.length;i++)
		{
			ans[i]=Integer.valueOf(cmd[i]);
		}
		return ans;
	}
	static long[] splitL() throws IOException
	{
		String[] cmd=br.readLine().split(" ");
		long[] ans=new long[cmd.length];
		for(int i=0;i<cmd.length;i++)
		{
			ans[i]=Long.valueOf(cmd[i]);
		}
		return ans;
	}
	static int n;
	static double[]arr;
	static double[][]dp;
	static double prob(int ind,int heads)
	{
		if(heads>n-heads)
			return 1;
		if(ind==n)
			return 0;
		if(dp[ind][heads]!=-1)
			return dp[ind][heads];
		double res=(1-arr[ind])*prob(ind+1,heads)+arr[ind]*prob(ind+1,heads+1);
		dp[ind][heads]=res;
		return res;
	}
	public static void main(String[] args) throws Exception{
		// TODO Auto-generated method stub
		br=new BufferedReader(new InputStreamReader(System.in));
		n=cin();
		String[] cmd=br.readLine().split(" ");
		arr=new double[n];
		for(int i=0;i<n;i++)
		{
			arr[i]=Double.valueOf(cmd[i]);
		}
		dp=new double[n+1][n+1];
		for(int i=0;i<=n;i++)
		{
			Arrays.fill(dp[i], (double)(-1));
		}
		System.out.println(prob(0,0));
	}
}